D. Calvo
GENERALIZED GEVREY CLASSES AND
MULTI-QUASI-HYPERBOLIC OPERATORS
Abstract. In this paper we consider generalized Gevrey classes defined in terms of Newton polyhedra. In such functional frame we prove a theorem of solvability of the Cauchy Problem for a class of partial differential op-erators, called multi-quasi-hyperbolic. We then give a result of regularity of the solution with respect to the space variables and finally analyze the regularity with respect to the time variable.
Introduction
It is well known from the Cauchy-Kovalevsky theorem that the Cauchy Problem for partial differential equations with constant coefficients or analytic coefficients, and an-alytic data admits a unique, anan-alytic solution.
But there are problems that are not C∞well-posed, i.e. starting with C∞data, there is not a C∞solution. In these cases it is natural to consider the behaviour of the operator in the Gevrey classes Gs,1 <s <∞(for definition and properties see for example Rodino [11]). Solvability of the Cauchy Problem in Gevrey spaces has been obtained for a class of partial differential operators with constant coefficients, the so called s-hyperbolic operators.
More precisely, we recall the following definition and the corresponding result, for the proof see Cattabriga [3], H¨ormander [9], Rodino [11].
DEFINITION1. Let’s consider partial differential operators inRn+1 =Rt×Rnx,
non-characteristic with respect of the t-hyperplanes, i.e. operators that can be written in the form:
(1) P(Dt,Dx)=Dtm+ m−1X
j=1
aj(Dx)Dtj
with or der(aj(Dx))≤m−j .
We say that P(D)is s-hyperbolic (with respect to the t variable), 1 <s <∞, if its symbol satisfies for a suitable C>0 the condition:
if λm+Pm−1j=0aj(ξ )λj =0 for(λ, ξ )∈Ct×Rnx, then ℑλ≥ −C(1+ |ξ| 1
s). In the caseℑλ≥ −C we say that P(D)is hyperbolic.
THEOREM1. Let P(D)be a differential operator inRt ×Rnx of the form (1) and
let P be s-hyperbolic with respect to t, with 1<s <∞. Let 1 <r < s and assume fk(x) ∈ Gr0(Rxn)for k = 0,1, . . . ,m−1. Then there exists a Gevrey function u ∈
Gr(Rn+1)satisfying the Cauchy Problem:
(2)
(
P(D)u =Dtmu+Pm−1j=0 aj(Dx)Dtju=0
Dktu(0,x)= fk(x) ∀x∈Rn,∀k=0,1, . . . ,m−1.
In the case P(D)is hyperbolic , we have the corresponding result of existence in the C∞class.
The previous Theorem 1 can be extended to operators with variable coefficients, for example we refer to the important contribution of Bronstein [2].
Here, remaining in the frame of constant coefficients, we want to extend the previous theorem in order to assure the solvability of the Cauchy Problem for a larger class of data.
To this end, we define generalized Gevrey classes GsP, 1 < s < ∞, based on a complete polyhedronP, following Zanghirati [13], Corli [6], and give equivalent
defi-nitions of these classes (for details see Section 1).
Let us observe that Gs ⊂ GsP. The classes GsP allow to express a precise result of regularity for the so called multi-quasi-elliptic equations, defined in terms of the norm
|ξ|Passociated toP, see Cattabriga [4], Hakobyan-Margaryan [8],
Boggiatto-Buzano-Rodino [1] and the subsequent Section 1.
We then introduce a class of differential operators with constant coefficients, modelled on a complete polyhedronP, that is natural to name as multi-quasi-hyperbolic opera-tors.
DEFINITION2. Let 1<s<∞and letP be a complete polyhedron. We say that a differential operator with constant coefficients inRn
x ×Rt of the form (1) is
multi-quasi-hyperbolic of order s with respect toPif there exists a constant C >0 such that
for(λ, ξ )∈C×Rnthe condition:
P(λ, ξ )=λm+ m−1X
j=0
aj(ξ )λj =0
implies:
ℑλ≥ −C|ξ| 1
s
P
where|ξ|Pis the weight associated toPas in Section 1.
The algebraic properties of the symbol of multi-quasi-hyperbolic operators will be studied in Section 2, where we shall also give some examples.
Since|ξ|P ≤const(1+ |ξ|), the previous definition implies s-hyperbolicity; therefore
the well-posedness of the Cauchy Problem in the larger classes GrP, r < s. More precisely, in Section 3 we will prove the following theorem:
THEOREM2. Let P(D)be a differential operator inRt×Rnxas in (1) and let P be
multi-quasi-hyperbolic of order s with respect to a complete polyhedronPinRn, with
1<s<∞. Let 1<r <s and assume fk ∈Gr0P(Rnx)for k=0,1, . . . ,m−1. Then
there exists u(t,·)∈GrP(Rn
x)for t ∈Rsatisfying the Cauchy Problem (2).
This gives a regularity of the solution u with respect to the space variables. To test the regularity with respect to the time variable, we need to define a new polyhedron
P′ that extends the polyhedronP toRn+1. We shall then be able to conclude u ∈
GrP′(Rn+1), see to Section 4 for details.
1. Complete polyhedra and generalized Gevrey classes
A convex polyhedronP inRnis the convex hull of a finite set of points inRn. There
is univocally determined byP a finite setV(P)of linearly independent points, called the set of vertices ofP, as the smallest set whose convex hull isP.
Moreover, ifPhas non-empty interior, there exists a finite set:
N(P)=N0(P)SN1(P) such that :
|ν| =1,∀ν∈N0(P) and
P = {z∈Rn|ν·z≥0,∀ν∈N
0(P)∧ν·z≤1,∀ν∈N1(P)}.
The boundary ofP is made of facesFνof equation: ν·z=0 ifν∈N0(P)
ν·z=1 ifν∈N1(P).
We now introduce a class of polyhedra that will be very useful in the following. DEFINITION3. A complete polyhedron is a convex polyhedronP ⊂Rn
+such that:
1. V(P)⊂Nn (i.e. all vertices have integer coordinates);
2. the origin(0,0, . . . ,0)belongs toP; 3. di m(P)=n;
4. N0(P) = {e1,e2, . . . ,en}, with ej = (0,0, . . . ,0,1j−t h,0, . . . ,0) ∈ Rn for
j=1, . . . ,n; 5. N1(P)⊂Rn
+.
We note that 5. means that the set: Q(x)= {y∈ Rn|0≤ y≤ x} ⊂P if x ∈ P
and if s belongs to a face ofPand r >s then r 6∈P.
PROPOSITION1. LetPbe a complete polyhedron inRnwith natural (or rational)
vertices sl =(sl1, . . . ,snl), l =1, . . . ,n(P), where n(P)is the number of the vertices
ofP, then:
1. for every j =1,2, . . . ,n, there is a vertex slj ofPsuch that: (0, . . . ,0,sljj,0, . . . ,0)=sljjej, s
lj
j =max
s∈Psj =: mj( P).
2. there is a finite non-empty setN1(P)⊂Qn
+\{0}such that:
P = \
ν∈N1(P)
{s∈Rn
+:ν·s≤1};
3. for every j =1, . . . ,n there is at least oneν∈N1(P)such that: mj =mj(P)=ν−1j ;
4. if s∈P, then:
|ξs| ≤ nX(P)
l=1 |ξsl|
ξs =
n
Y
j=1 ξsjj
.
The proof is trivial and we need only to point out that 4. is a consequence of the following lemma, for whose proof we refer to Boggiatto-Buzano-Rodino [1], Lemma 1.1.
LEMMA 1. Given a subset A ⊂ (R+0)n and a linear convex combinationβ =
P
α∈Acαα, then for any x∈(R+0)nthe following inequality is satisfied:
(3) xβ ≤X
α∈A
cαxα
We now give some notations related to a complete polyhedronP.
Let’s denote by L(P)the cardinality of the smallest set N1(P)that satisfies 2. of Proposition 1.
We denote:
Fν(P)= {s∈P :ν·s=1}, ∀ν∈N1(P) a face ofP;
F=S
ν∈N1(P)Fν(P)the boundary ofP;
V(P)the set of vertices ofP;
δP = {s∈Rn
+:δ−1s∈P}, δ >0;
k(s,P)=inf{t >0 : t−1s∈P} =maxν∈N1(P)ν·s, s∈Rn+.
Now letPbe a complete polyhedron, we say: µj(P)=maxν∈N1(P)ν−1j ;
µ=µ(P)=maxj=1,...,nµj the formal order ofP;
µ(1)(P)=maxγ∈V(P)|γ| the maximum order ofP;
q(P)=µµ(P) 1(P), . . . ,
µ(P) µn(P)
;
|ξ|P =(Ps∈V(P)ξ2s)
1
2µ,∀ξ ∈Rn the weight ofξassociated to the polyhedronP.
Considering a polynomial with complex coefficients, we can regard it as the symbol of a differential operator, and associate a polyhedron to it, as in the following.
DEFINITION 4. Let P(D) = P|α|≤mcαDα, cα ∈ Cbe a differential operator with complex coefficients inRnand P(ξ ) =P
|α|≤mcαξα, ξ ∈ Rnits characteristic polynomial. The Newton polyhedron or characteristic polyhedron associated to P(D) is the convex hull of the set:
{0}[{α∈Zn
+: cα 6=0}.
There follow some examples of Newton polyhedra related to differential operators: 1. If P(ξ )is an elliptic operator of order m, then its Newton polyhedron is complete and is the polyhedron of vertices{0,mej, j =1, . . . ,n}and so:P = {ξ ∈Rn: ξ ≥0, Pni=1ξi ≤m}.
The setN1(P)is reduced to a point: ν=m−1Pmj=1ej =(m−1, . . . ,m−1).
mj(P)=µj(P)=µ(0)(P)=µ(1)(P)=µ(P)=m, j=1,2, . . . ,n;
q(P)=(1,1, . . . ,1);
k(s,P)=m−1|s| =m−1Pnj=1sj, s∈Rn+.
2. If P(ξ )is a quasi-elliptic polynomial of order m (see for example H¨ormander [9], Rodino [11], Zanghirati [12]), its characteristic polyhedronP is complete and has vertices{0,mjej, j =1, . . . ,n}where mj =mj(P)are fixed integers.
The setN1(P)is again reduced to a point: ν=Pnj=1m−1j ej.
P = {ξ ∈Rn:ξ ≥0,Pn
j=1m−1j ξj ≤1}; µj(P)=mj, j =1, . . . ,n;
µ(0)(P)=minj=1,...,nmj;
µ(P)=µ(1)(P)=maxj=1,...,nmj =m;
q(P)=(m1m , . . . ,mm n);
k(s,P)=µ(P)−1q·s, s∈Rn +.
In this case the unique face ofPis defined by the equation:
1 m1
x1+. . .+
1 mn
xn=1.
We note in general that s belongs to the boundary of k(s,P)Pand k(s,P)is univocally determined for complete polyhedra.
k(s,P)satisfies the following inequality that will be very useful in the following:
(4) |s|
µ(1) ≤k(s,P)≤ |
|s|
µ(0) ≤ |s|, ∀s∈R
We remember (see [1]) that the polyhedron of a hypoelliptic polynomial is complete, but the converse is not true in general.
We now introduce a class of generalized Gevrey functions associated to a complete polyhedron, as in Corli[6], Zanghirati [13].
They can be regarded as a particular case of inhomogeneous Gevrey classes with weight λ(ξ )= |ξ|P, in the sense of the definition of Liess-Rodino [11], and can be expressed
also by means of the derivatives of u.
Following Corli [6] we give the following definition:
DEFINITION5. LetPbe a complete polyhedron inRn. Letbe an open set inRn
and s∈R, s>1. We denote by GsP()the set of all u∈C∞()such that:
(5) ∀K ⊂⊂, ∃C >0 :
|Dαu(x)| ≤C|α|+1(µk(α,P))sµk(α,P), ∀α∈Zn
+, ∀x∈K.
We also define:
Gs0P()=GsP()∩C0∞().
The space GsP()can be endowed with a natural topology. Namely, we denote by C∞(P,s,K,C)the space of funcions u∈C∞()such that:
(6) suppu⊂K
kukK,C =supα∈Zn
+supx∈KC
−|α|(µk(α,P))−sµk(α,P)|Dαu(x)|<∞ With such a norm, C∞(P,s,K,C)is a Banach space. Then:
GsP()= \ K⊂⊂
[
C>0
C∞(P,s,K,C)
endowed with the topology of projective limit of inductive limit.
REMARK1. IfP is the Newton polyhedron of an elliptic operator, then GsP() coincides with Gs(), the set of the standard s-Gevrey functions in.
REMARK2. IfPis the Newton polyhedron of a quasi-elliptic operator, then:
GsP()=Gsq(), where q=
m m1
, . . . , m mn
the set of the anisotropic Gevrey functions, for definition see H¨ormander [9], Rodino [11], Zanghirati [12].
REMARK3. We have the following inclusion: Gs
µ
µ(1) ⊂GsP
⊂Gs
µ
µ(0), ∀s>1, ∀P
We give now equivalent definitions of generalized Gevrey classes.
The arguments are similar to those in Corli [6], Zanghirati [13], but simpler, since for our purposes we need to consider only classes for s >1; to be definite, we prefer to give here self-contained proofs.
LetP be a complete polyhedron inRnand let K be a compact set inRn.
DEFINITION6. Ifν∈N1(P), let:
C(ν)= {α∈Zn+: k(α,P)=α·ν}. C(ν)is a cone ofZn
+and C(ν)
TF
=Fν. This means that k(α,P)−1α∈Fν.
LEMMA2. Let s>1, there is a functionχ ∈C0∞(Rn)such that: χ (x)=1, x∈ K,
|Dαχ| ≤C(C Nsµ)α·ν, ifα·ν≤N, ∀N =1,2, . . . , ∀ν∈N1(P). (7)
Proof. Every u∈ Gs0(Rn)satisfies the conditions 7. In fact, every u ∈ Gs
0(Rn)
satis-fies:
|Dαu(x)| ≤CC|α||α|s|α|≤CC|α|Ns|α| if|α| ≤N. In fact, as 0< νj ≤1,∀j=1, . . . ,n,∀ν∈N1(P)andα·ν≤ |α|, we get:
|α| ≤α·νmax(νj)−1=α·νµ |α| ≤N ⇒α·ν≤N.
So, taking R=Cµµsµ, we obtain:
|Dαu(x)| ≤C(R Nsµ)α·ν, ∀ν ∈N1(P), ∀α:α·ν≤N.
Then we can proceed as in the C∞case to constructχ ∈ Gs0(Rn)such thatχ ≡1 in
K .
LEMMA 3. With the previous notations, if u ∈ GsP(Rn), then taking χ as in
Lemma 2, we obtain the estimate: (8) |χcu(ξ )| ≤C
C Ns
|ξ|P+Ns
µN
N =1,2, . . . Proof. By Leibniz formula we can write:
|Dα(χu)| ≤X
β≤α α
β
|Dα−βχ||Dβu|
Let’s choose anyβ ≤α, thenβ ∈C(ν)for someν∈N1(P)(not necessarily unique)
and for thatνwe get:
sup
x∈suppχ|D
and by Lemma 2:
sup
x∈suppχ|D
α−βχ (x)| ≤C(C Nsµ)(α−β)·ν ifα·ν≤ N, N=1,2, . . .
So we get: sup
x∈suppχ|D
α−βχ (x)||Dβu(x)| ≤C(C Nsµ)(α−β)·νC|α−β|+1
1 (µk(β,P)) sµk(β,P)
≤C(C Nsµ)(α−β)·νC1(C2µk(β,P))sµk(β,P)
as|β| ≤µk(β,P), taking C2=C 1
s
1.
But we have supposed that k(β,P)=β·νandβ ≤α, moreoverα·ν ≤ N implies β·ν≤ N . We now proceed to estimate:
sup
x∈suppχ|D
α−βχ (x)||Dβu(x)| ≤C′(C Nsµ)(α−β)·ν(C
1Nsµ)β·ν ≤C′(C′′Nsµ)α·ν
∀α, ifα·ν ≤ N, ∀ν ∈N1(P), N =1,2, . . .. Taking C′′ =max{C,C1}, using the linearity of scalar product and observing that:
(α−β)·ν+β·ν=α·ν,
k(α,P)=max{α·ν, ν∈N1(P)}
we get the inequality:
|Dα(χu)| ≤C′(C′′Ns)µk(α,P). On the other hand we have:
|D\α(χu)| =| Z
e−i x·ξDα(χu)| ≤
Z
suppχ|D
α(χu)| ≤C sup suppχ|D
α(χu)|
asχhas compact support. Using the properties of the Fourier transform we conclude:
|D\α(χu)| = |ξα c
χu| ≤C sup suppχ|D
α(χu)| ≤C(C Ns)µk(α,P).
Let nowα=vN , for anyv∈V(P), the set of vertices ofP, summing up the previous inequalities forα=0, α=vN,∀v ∈V(P), we obtain:
|χcu(ξ )|NsµN+ X
v∈V(P)
Using the following inequality:
where n(P)denotes the number of vertices ofPdifferent from the origin. So we can conclude that:
|χcu(ξ )| ≤ C(C N
REMARK4. The previous Theorem 3 admits the more general formulation: Let K ⊂⊂,u∈D′(), then u is of class GsP
in a neighborhood U of K if and only if there isv∈E′()orv∈S′(Rn), v=u in U such thatvˆsatisfies the estimate (9).
The proof is analogous to that of Theorem 3.
Proof. Proof of necessity: Let u∈GsP in the set{x :|x−x0| ≤3r}, 0<r≤1, χ as in Lemma 3, with K = {x :|x−x0| ≤r}and suppχ ⊂ {x :|x−x0| ≤2r}. Then the functionv=χu satisfies conditions 1.,2. of the theorem.
Proof of sufficiency:
Letv ∈ E′()satisfy the conditions 1.,2.. Then there are two constants M
Now we use the property:
and hence by Lemma 1:
|ξα| =
Now, splitting the integral into the two regions:
|ξ|P <Ns, |ξ|P >Ns
The first integral is limited for all N and the second converges for large N , namely we set N =k(α,P)+R for R depending only onPand M. Then:
|Dαu(x)| ≤C′(C′µk(α,P)+R)sµ(k(α,P)+R) implies:
|Dαu(x)| ≤C|α|+1(µk(α,P))sµk(α,P).
THEOREM4. 1. Let u∈G0sP(Rn), then there exist two constants C>0, ǫ >
In the proof of Theorem 4 we shall use the following lemma: LEMMA4. The estimates:
(13) | ˆu(ξ )| ≤C
are equivalent to the following:
(14) | ˆu(ξ )| ≤CN+1N !|ξ|−
N s
P , N =1,2, . . .
for suitable different constants C>0 independent of N .
The proof of the lemma is trivial and based on the inequalities: N ! ≤NN, ∀N =
Then by the previous lemma, u satisfies for a suitable constant C′>0:
and hence: So we obtain for a suitable constant C>0:
| ˆu(ξ )| ≤Cex p(−ǫsµ|ξ| Hence, by expanding the exponential into Taylor series we have:
∞
and hence for a new constant C>0:
| ˆu(ξ )| ≤C′
For any complete polyhedronP we define the corresponding class of hyperbolic operators, according to Definition 2. For short, we denote multi-quasi-hyperbolic operators of order s with respect toP by(s,P)-hyperbolic.
Obviously, if P(D)is multi-quasi-hyperbolic of order s >1 with respect toP, P(D) is also multi-quasi-hyperbolic of order r, ∀r, 1<r<s with respect toP.
PROPOSITION 2. If P(D) is (s,P)-hyperbolic for 1 < s < ∞, then for any
(λ, ξ )∈C×Rnsuch that P(λ, ξ )=0, there is C>0 such that:
(15) |ℑλ| ≤C|ξ|
1
s
P.
Proof. The coefficient ofλm−1 in P(λ, ξ ) is a linear function ofξ. If the zeros of P(λ, ξ )are denoted byλj, it follows thatPmj=1λj is a linear function ofξ. Then also
Pm
j=1ℑλj is a linear combination ofξ, and if P(D)is(s,P)-hyperbolic, then: m
X
j=0
ℑλj ≥ −mC|ξ| 1
s
P
impliesPmj=0ℑλj = C0 for a suitable constant C0. So we obtain for allλk root of
P(λ, ξ ):
ℑλk =C0−
X
j6=k
ℑλj ≤C0+C(m−1)|ξ| 1
s
P≤C′|ξ|
1
s
P .
That completes the inequality:
|ℑλk| ≤C|ξ| 1
s
P
for all rootsλk of P(λ, ξ ).
PROPOSITION3. If P(D)is(s,P)-hyperbolic for 1 <s < ∞, then the princi-pal part Pm(D)of P(D) is hyperbolic, i.e. the homogeneous polynomial Pm(λ, ξ )
satisfies:
(16) Pm(λ, ξ )=0 (λ, ξ )∈C×Rn ⇒ ℑλ=0.
Proof. Takingσ >0, λ∈C, ξ ∈Rn, we get:
Pm(λ, ξ )= lim
σ→∞P(σ λ, σ ξ )·σ −m
From Proposition 2 the zeros of P(σ λ, σ ξ )must satisfy:
|ℑλk| ≤C |σ ξ|
1
s
P
σ
So forσ → ∞,ℑλ=0 for all the rootsλ∈Cof Pm(λ, ξ ), that is Pm(D)is hyperbolic.
PROPOSITION4. For a differential operator Pm(D)associated to an homogeneous
polynomial Pm(λ, ξ ), the notion of hyperbolicity and(s,P)-hyperbolicity coincide.
PROPOSITION5. Let P(D)be a differential operator of the form:
(17) P(D)=Pm(D)+
m−1X
j=0
aj(Dx)Dtj
with homogeneous principal part:
(18) Pm(D)=Dtm+
m−1X
j=0
bj(Dx)Dtj,
with:
or der(bj(Dx))=m− j
or der(aj(Dx))≤m−j−1
and assume:
(19) Pm(λ, ξ )=0 forλ∈C, ξ∈R
nimpliesℑλ=0 |aj(ξ )| ≤C|ξ|m−1P (1+ |ξ|)−j for j=1, . . . ,m−1
(for a C>0).
Then P(D)is(m−1m ,P)hyperbolic.
Proof. By definition, the terms aj(Dx), bj(Dx)satisfy for a suitable C >0:
(20) |bj(ξ )| ≤C|ξ|
m−j
|aj(ξ )| ≤C|ξ|m−j−1.
In the region{ǫ|λ| > |ξ|}(forǫ > 0 sufficiently small), the following inequality is satisfied:
|P(λ, ξ )−λm| ≤C
m−1X
j=0
|ξ|m−j|λ|j <λ m
2 that implies:
|P(λ, ξ )|>λ m
2
and consequently P(λ, ξ )can’t have roots in this region and the roots must so satisfy forǫ >0:
(21) |λ| ≤ǫ−1|ξ|.
On the other hand, for(λ, ξ )such that P(λ, ξ )=0:
Pm(λ, ξ )= −(P−Pm)(λ, ξ )= − m−1X
j=0
In view of the estimates (21) and (19), we obtain:
|Pm(λ, ξ )| ≤ m−1X
j=0
|aj(ξ )||λ|j ≤C m−1X
j=0
|ξ|m−1P (1+ |ξ|)−j|λ|j
≤C′
m−1X
j=0
|ξ|m−1P (1+ |ξ|)−j|ξ|j ≤C′′|ξ|m−1P
In view of the hyperbolicity of Pm we can write:
Pm(λ, ξ )= m
Y
j=0
(λ−λj), λj ∈R.
Hence:
|ℑλ|m ≤ |Pm(λ, ξ )| ≤C′′|ξ|m−1P
|ℑλ| ≤C′′′|ξ|
m−1
m
P
i.e. P(D)is(m−1m ,P)hyperbolic.
PROPOSITION6. Any differential operator P(D)=Dmt +Pm−1j=0 aj(Dx)Dtj
sat-isfying the condition:
(22) |aj(ξ )| ≤C|ξ|m−P j−1 j =0,1, . . . ,m−1, for C >0
is(m−1m ,P)hyperbolic.
We note that the principal part is only Dmt and is obviously hyperbolic; Proposition 6 states that in this particular case we may replace (19) with the weaker assumption (22).
Proof. By the estimates (22) we have:
|P(λ, ξ )−λm| ≤C
m−1X
j=0
|ξ|m−jP −1|λ|j <|λ| m
2 in the region{(λ, ξ )∈C×Rn:|ξ|
P < ǫ|λ|}for a sufficiently smallǫ >0.
Consequently|P(λ, ξ )|> |λ2|m and P(λ, ξ )can’t have roots in this region and so they must satisfy:
(23) |λ| ≤ǫ−1|ξ|P.
For(λ, ξ )such that P(λ, ξ )=0, we write:
λm= −(P(λ, ξ )−λm)= − m−1X
j=0
In view of the estimate (23) forλand (22) for aj(ξ ):
|λ|m ≤C′
m−1X
j=0
|ξ|m−jP −1|λ|j ≤C′′|ξ|m−1P
Hence:
|ℑλ|m≤C′′|ξ|m−1P |ℑλ| ≤C′′′|ξ|
m−1
m
P
i.e. P(D)is(m−1m ,P)hyperbolic.
REMARK 5. A more general version of Proposition 6 is easily obtained by as-suming as in Proposition 5 that P(D) has hyperbolic homogeneous principal part Pm(D)=Pm−1j=0 bj(Dx)Dtj with:
(24) |bj(ξ )| ≤C|ξ|Pm−j, j =0,1, . . . ,m−1
and keeping condition (22) for the lower order terms.
Observe however that (24) implies bj(ξ )≡0, but in the quasi-homogeneous case.
There follow some examples of multi-quasi-hyperbolic operators, that follow from the previous propositions.
1. If P(D)is a differential operator inRnwith symbol P(ξ )and Newton
polyhe-dronP of formal orderµ, then the differential operator inRn+1:
Q(D)=Dmt +P(Dx),
with m> µ, is multi-quasi-hyperbolic of ordermµ with respect toP. In fact, the roots of the symbol of Q(D)satisfy:
|ℑλ| ≤C|ξ|
µ
m
P.
2. A particular case of Proposition 5 is the following:
ifPis the polyhedron inR2of vertices(0,0), (0,2), (1,0), thenµ=2 and the
following operator:
P(Dx,Dt)=P3(Dx,Dt)+C1D2x2+C2Dx1 +C3Dx2+C4Dt+C5
where P3(Dx,Dt) is an hyperbolic homogeneous operator of order 3 and
3. Another particular case of Proposition 5 is the following:
ifP is the polyhedron inR2 of vertices(0,0), (0,3), (1,2), (2,0), then the
formal orderµ=4 and the operator of order 4:
P(Dt,Dx)=P4(Dt,Dx)+c1Dx22+c2Dx1Dx2 +c3Dx2Dt +c4Dx1 +c5Dx2 +c6Dt+c7
where P4(Dx,Dt) is an hyperbolic homogeneous operator of order 4 and
C1, ...,C7∈C, is multi-quasi-hyperbolic of order43 with respect toP.
4. Let P(D)be a differential operator inRnwith symbol P(ξ ), then we consider
the differential operator inRn+1:
Q(D)=(Dt2+ △x)m−P(Dx)
with or der P(D) <2m.
The roots of the symbol of Q(D)satisfy:
(λ2− |ξ|2)m−P(ξ )=0
and then, denoting by P(ξ )m1 the generic m−th root of P(ξ ):
ℑλ= |ξ2+P(ξ )m1|12senθ
where forθ >0:
tg2θ = ℑ(ξ
2+P(ξ )m1) ℜ(ξ2+P(ξ )m1)
≤ |P(ξ )| 1
m
|ξ|2 .
We consider the first term of the Taylor expansion to estimate senθ:
senθ≤C|P(ξ )|
1
m 2|ξ|2 |ℑλ| ≤CP(ξ )
1
m
|ξ|
LetP′be a given complete polyhedron. If for someρ <1 we have: |P(ξ )|m1 ≤C|ξ|ρ
P′|ξ| i.e.
|P(ξ )| ≤C|ξ|ρPm′|ξ|m
(25)
then Q(D)is multi-quasi-hyperbolic of order ρ1 with respect toP′. If we
3. Proof of Theorem 2
Now we prove Theorem 2.
Proof. We try to satisfy the Cauchy Problem: (
P(D)u =Dmt u+Pk=0m−1ak(Dx)Dktu=0
Dktu(0,x)= fk(x) ∀x ∈Rn,∀k=0,1, . . . ,m−1
by a function u(t,x)such that u(t,x)∈S(Rn
x)for any fixed t∈R.
We apply partial Fourier transform with respect to x , considering t as a parameter, so the Cauchy Problem admits the following equivalent formulation:
(26)
(
P(Dt, ξ )uˆ=Dtmuˆ+
Pm−1
k=0 ak(ξ )Dtjuˆ=0
Dktuˆ(0, ξ )= ˆfk(ξ ) ∀ξ ∈Rn, k=0,1, . . . ,m−1.
This makes sense as fkhave compact support,∀k=0,1, . . . ,m−1 and u ∈S(Rn),∀t
fixed.
Now we consider the Cauchy Problem (26) as an ordinary differential problem in t, depending on the parameterξ. A solution to problem (26) is given by:
(27) uˆ(t, ξ )=
m−1X
j=0 ˆ
fj(ξ )Fj(t, ξ ),
where Fj(t, ξ ), j = 0,1. . . . ,m−1, satisfy the ordinary Cauchy Problem on t
de-pending on the parameterξ ∈Rn:
(28)
(
P(Dt, ξ )Fj =0
DktFj(0, ξ )=δj k k=0,1, . . . ,m−1
whereδj kdenote the Kronecker delta.
The solution of (28) exists and is unique by the Cauchy theorem for ordinary dif-ferential equations, and the functionu defined in (27) gives indeed a solution to theˆ
Cauchy Problem (26), as is easy to check. Now we want to estimate|Dαxu(t,x)|or, equivalently,| ˆu(t, ξ )|to obtain generalized Gevrey estimates with respect to the space variables.
By assumption fˆj(ξ ) ∈ Gr0P(Rn), so, in view of Theorem 4,(1), there are constants ǫj,Cj >0(j =0,1, . . . ,m−1)such that for everyξ ∈Rn:
| ˆfj(ξ )| ≤Cjexp(−ǫj|ξ| 1
r
P)≤C exp(−ǫ|ξ|
1
r
P),
taking:
C =max{Cj, j=0, . . . ,m−1}, ǫ=max{ǫj, j=0, . . . ,m−1}.
To estimate Fj we use the following lemma (for the proof see for example
LEMMA 5. Let L(D)= Dm +Pm−1j=0 ajDj be an ordinary differential operator
with constant coefficients aj ∈C. Write3= {λ∈C: L(λ)=0}and assume:
max
λ∈3|λ| ≤A, max
λ∈3|ℑλ| ≤B forλ∈3. (29)
Then the solutionsvj(t), j=0,1, . . . ,m−1 of the Cauchy Problems:
(30)
(
L(D)vj =0
(Dkvj)(0)=δj k, k=0, . . . ,m−1
satisfy the following estimates:
|DNvj(t)| ≤2m(A+1)N+m+1e(B+1)|t|,
N=0,1, . . . , t ∈R. (31)
We now apply the estimates of Lemma 5 for N = 0 to the functions Fj(t, ξ )in
(28), j =0,1, . . . ,m−1, takingξ as a parameter. If P(D)is(s,P)-hyperbolic, then
∃C′>0 such that the roots of P(λ)satisfy:
|ℑλ| ≤C′|ξ| 1
s
P,
consequently we may take B=C′|ξ| 1
s
P.
Now we determine A. Let’s consider the characteristic polynomial of P:
P(λ, ξ )=λm+ m−1X
j=0
aj(ξ )λj
where aj(ξ )is a polynomial of degree at most equal to m− j . So there are constants
Cj such that:
|aj(ξ )| ≤Cj(1+ |ξ|)m−j.
It follows easily that forǫ >0 sufficiently small the zeros of P(λ, ξ )cannot belong to the region{(1+ |ξ|) < ǫ|λ|}and must necessarily satisfy:
(32) |λ| ≤ǫ−1(1+ |ξ|) .
So we can take:
(33) A=ǫ−1(1+ |ξ|)
and estimate for a suitable C>0:
(34) |Fj(t, ξ )| ≤(ǫ−1(1+ |ξ|)+1))m+1C exp(C(|t| +1)|ξ| 1
s
By summing up the estimates for fˆj,Fj we get the following estimates foru:ˆ
| ˆu(t, ξ )| ≤ m−1X
j=0
| ˆfj(ξ )||Fj(t, ξ )|
≤C
m−1X
j=0
exp(−ǫ|ξ| 1
r
P)exp(C(1+ |t|)|ξ|
1
s
P) .
(35)
By assumption, r <s, and so1r > 1s implies that:
lim
|ξ|→+∞ |ξ|
1
s
P
|ξ| 1
r
P
=0
Then there exist positive constants C1′ =C1′(|t|), C′2=C2′(|t|)such that:
C(1+ |t|)|ξ| 1
s
P−ǫ|ξ|
1
r
P ≤ −C1′|ξ| 1
r
P+C2′ .
Hence we get the following estimate foru:ˆ
| ˆu(t, ξ )| ≤C′′exp(−C1′|ξ| 1
r
P).
So we have obtained that u∈GrPfor any t∈Rin view of Theorem 4,2). We observe that the constants C′1,C′′may depend on t, but are locally bounded, for|t| ≤T,∀T > 0.
REMARK6. We have supposed that r >s to get the result of regularity. In the case r =s, the regularity is only local in time, as evident from the previous computations.
4. Regularity with respect to the time variable
We know that the solution of the Cauchy Problem is in C∞([−T,T ],GrP(Rn)),∀T >
0 ; now we will discuss its regularity with respect to the time variable in generalized Gevrey classes. To do so, it is necessary to extend the polyhedron to(n+1)variables, that is possible by means of the following proposition.
PROPOSITION7. Given a complete polyhedronPinRn, we defineP′as the convex
hull inRn+1of the vertices ofP plus the vector(µ
0,0, . . . ,0)withµ0 ∈ Q+, 0 < µ0 ≤ µ, cf. figure. ThenP′ is a complete polyhedron inRn+1with the same formal
✠
✲ ✻
❅ ❅
❉ ❉ ❉❉
✁✁ ✁✁
✁✁ ✁✁
✁✁
✪✪ ✪✪
✪✪ ✪✪✪
✑✑ ✑✑
✑✑ ✑✑
P′ P
µ0
The proof is trivial and follows immediately from the definition of complete poly-hedra and of formal order. Let’s observe that it is not possible to constructP′ with
a smaller formal order. Of course, one could take more than one additional vertex to buildP′; the construction in Proposition 7 represents the cheapest procedure, which
could be easily iterated to extendP to(n+m)dimensions,∀m.
DEFINITION7. We callP′in Proposition 7 an extension ofPinRn+1.
If the further vertex has coordinates(µ,0, . . . ,0)withµdenoting the formal order of
P, we say thatP′is the maximal extension ofP inRn+1.
PROPOSITION8. LetPbe a complete polyhedron inRnand letP′be an extension
ofP toRn+1by the additional vertex (µ
0,0, . . . ,0). Then for any α′ = (α0, α) ∈
Rn+1+ :
(36) k(α′,P′)=k(α,P)+α0 µ0
=k(α,P)+k(α0,R)
whereRdenotes the one-dimensional polyhedron [0, µ0] inR.
Proof. Writingα0= µα0
0µ0, we now compute k(α
′,P′)=k((α
0, α),P′).
Let us write:
α=k(α,P) nX(P)
i=1
tisi
si ∈V(P), nX(P)
i=1
ti =1 0≤ti ≤1, i =1, . . . ,n(P);
(α0,0, . . . ,0)= α0 µ0
(µ0,0, . . . ,0)=t0k(α0,R)s0,
t0=1, k(α0,R)= α0 µ0
We want to find k(α′,P′)such that:
On the other hand we have:
(α0, α)=
µ0 is univocally determined as s0is orthogonal to
P.
We will prove first a theorem of regularity of the Cauchy Problem with respect to the time variable in the particular case when the coefficients aj(ξ )satisfy the condition:
(37) |aj(ξ )| ≤C|ξ|Pm−j j =0,1, . . . ,m−1
and then a theorem for general aj(ξ )which requires a further discussion on the relation
between the euclidean norm inRn+1and the weight associated to the polyhedron.
THEOREM 5. Under the assumptions of Theorem 2, if (37) is satisfied, then the solution u of the Cauchy Problem (2) is of class GrP′
(Rn+1)whereP′ denotes the
maximal extension ofP toRn+1.
Proof. We have to test the regularity of u with respect to the time variable. Let us go back to the proof of Theorem 2. From (27) we have:
|DtNuˆ(t, ξ )| ≤ m−1X
j=0
By Lemma 5 , we can estimate:
(38) |DtNFj(t, ξ )| ≤2m(A+1)N+m+1exp((B+1)|t|).
By the hypothesis of multi-quasi-hyperbolicity for P(D), as before we may take:
B=C1|ξ| 1
s
P.
To determine A we use the hypothesis (37) that implies:
|P(λ, ξ )−λm| = | m−1X
j=0
aj(ξ )λj| ≤C m−1X
j=0
|ξ|m−jP |λ|j < |λ| m
2
in the region{(λ, ξ )∈C×Rn:|ξ|
P < ǫ|λ|}for a sufficiently smallǫ >0.
Consequently,|P(λ, ξ )|>|λ2|m and the zeros of P(λ, ξ )can’t be in this region, so they must satisfy:
|λ| ≤C|ξ|P , for C >0.
So we can take A=C|ξ|Pand estimate:
|DtNFj(t, ξ )| ≤2m(C|ξ|P)N+m+1exp(C0(|t| +1)|ξ| 1
s
P)
≤C(C′|ξ|P)Nexp(C1(|t| +1)|ξ| 1
s
P) .
Hence:
|DtNuˆ(t, ξ )| ≤ m−1X
j=0
| ˆfj(ξ )||DtNFj(t, ξ )|
≤C(C′|ξ|P)Nexp{(C1(|t| +1)|ξ| 1
s
P)−ǫ|ξ|
1
r
P}.
Arguing as in the proof of Theorem 2, we obtain for a suitableǫ1>0:
(39) |DtNuˆ(t, ξ )| ≤C(C′|ξ|P)Nexp(−ǫ1|ξ|
1
r
P).
Now we pass to consider the Fourier antitransform ofu with respect to the space vari-ˆ
ables:
u(t,x)=F−1
and estimate forα′=(α0, α): |Dα′u(t,x)| = |Dα0
t Dαxu(t,x)| = |D
α0
t F−1(ξαuˆ(t, ξ ))| = |Dα0
t [(2π )−n
Z
Rn
ei xξξαuˆ(t, ξ )dξ]| ≤(2π )−n
Z
Rn
|ξα||Dα0
t uˆ(t, ξ )|dξ ≤C(2π )−n
Z
Rn
|ξ|µPk(α,P)(C′|ξ|P)α0exp(−ǫ1|ξ|
1
r
P)dξ
≤C(2π )−n Z
Rn
(C|ξ|P)µ(k(α,P)+α0)exp(−ǫ1|ξ| 1
r
P)dξ
≤C|α|+α0+1(µk(α,P)+α0)r(µk(α,P)+α0)
where we have used (11) and we have followed the arguments of the proof of Theorem 4 (2).
LettingP′be the maximal extension ofPinRn+1, by Proposition 8:
k(α′,P′)=k(α,P)+α0 µ .
So we can conclude that for a suitable constant C>0:
(40) |Dα′u(t,x)| ≤C|α′|+1(µk(α′,P′))rµk(α′,P′), ∀α′ ∈Zn+1+ that means u∈GrP′(Rn+1)as we wanted to prove.
REMARK7. If r =s the regularity is only local in time and u is of class GsP′only in the set|t|<Cǫ
1 as to satisfy condition (39).
THEOREM 6. Under the assumptions of Theorem 2 the solution u of the Cauchy Problem is of class GrP′(Rn+1), where P′ is the extension of P toRn+1 obtained
adding the vertex:
s0=(µ0,0, . . . ,0), µ0=µ(0),
µ(0)=µ(0)(P)=min{mj : mjej ∈V(P), j =1, . . . ,n} =minγ∈V(P)\{0}|γ|.
Sinceµ0< µbut in the elliptic case, the present result of regularity is weaker than
the one expressed by Theorem 5 under the additional assumption (37). Proof. We proceed as in the proof of Theorem 5 to estimate:
(41) |DtNuˆ(t, ξ )| ≤ m−1X
j=0
From Lemma 5 we have:
by the hypothesis of multi-quasi-hyperbolicity, and now: A=C′(1+ |ξ|) . So arguing as in the previous proof we can estimate:
|Dα0
and passing to the Fourier antitransform with respect toξ:
|Dα′u(t,x)| = |Dα0
and therefore we can get the estimate:
|Dα′u(t,x)| ≤(2π )−nCα0+1
REMARK8. Analogously to Theorem 5 if r =s the regularity is only local in time and u is of class GsP′(Rn)only in the set|t|< ǫ, withǫ >0 depending on the initial
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AMS Subject Classification: 35E15, 35L30, 46Fxx, 35B65.
Daniela CALVO
Dipartimento di Matematica Universit`a di Torino via Carlo Alberto, 10 10123 Torino, ITALIA e-mail:[email protected]
